An Alternative Direct Simulation of Minsky Machines into Classical Bunched Logics via Group Semantics

Recently, Brotherston & Kanovich, and independently Larchey-Wendling & Galmiche, proved the undecidability of the bunched implication logic BBI. Moreover, Brotherston & Kanovich also proved the undecidability of the related logic CBI, as well as its neighbours. All of the above results are based on encodings of two-counter Minsky machines, but are derived using different techniques. Here, we show that the technique of Larchey-Wendling & Galmiche can also be extended, via group Kripke semantics, to prove the undecidability of CBI. Hence, we propose an alternative direct simulation of Minsky machines into both BBI and CBI. We identify a fragment called elementary Boolean BI (eBBI) which is common to the BBI/CBI families of logics and we show that the problem of Minsky machine acceptance can be encoded into eBBI. The soundness of the encoding is derived from the soundness of a goal directed sequent calculus designed for eBBI. The faithfulness of the encoding is obtained from a Kripke model based on the free commutative group Z.